CS344: Introduction to Artificial Intelligence

1. Lecture: 22-23 Herbrand’s Theorem Proving satisfiability of logic formulae using semantic trees (from Symbolic logic and mechanical theorem proving) By Raunak Pilani Under the guidance of Prof. P. Bhattacharyya CS344: Introduction to Artificial Intelligence

2. Basic Definitions • Interpretation: Assignment of meaning to the symbols of a language • Interpretations of Predicate logic requires defining: • Domain Of Discourse (D), which is a set of individuals that the quantifiers will range over • Mappings for every constant, n-ary function and n-ary predicate to elements, n-ary functions (DnD) and n-ary relations on D, respectively

3. Basic Definitions (contd.) • Satisfiability(Consistency) • A formula G is satisfiableiff there exists an interpretation I such that G is evaluated to “T” (True) in I • I is then called a model of G and is said to satisfy G • Unsatisfiability (Inconsistency) • G is inconsistent iff there exists no interpretation that satisfies G

4. Need for the theorem • Proving satisfiability of a formula is better achieved by proving the unsatisfiability of its negation • Proving unsatisfiability over a large set of interpretations is resource intensive • Herbrands Theorem reduces the number of interpretations that need to be checked • Plays a fundamental role in Automated Theorem Proving

5. Skolem Standard Form • Logic formulae need to first be converted to the Skolem Standard Form, which leaves the formula in the form of a set of clauses • This is done in three steps • Convert to Prenex Form • Convert to CNF (Conjunctive Normal Form) • Eliminate existential Quanitifiers using Skolem functions

6. Step 1: Converting to Prenex Form • Involves bringing all quantifiers to the beginning of the formula • (Qi xi) (M), i=1, 2..., n Where, - Qi is either V (Universal Quantifier) or Ǝ (Existential Quanitifier) and is called the prefix - M contains no Quantifiers and is called the matrix

7. Example

8. Step 2: Converting to CNF • Remove and • Apply De Morgan’s laws • Apply Distributive laws • Apply Commutative as well as Associative laws

9. Example

10. Step 3: Skolemization • Consider the formula, (Q1 x1)… (Qn xn)M • If an existential quantifier, Qr is not preceded by any universal quantifier, then • xr in M can be replaced by any constant c and Qr can be removed • Otherwise, if there are ‘m’ universal quantifiers before Qr , then • An m-place function f(p1 , p2 ,… , pm) can replace xr where p1 , p2 ,… , pm are the variables that have been universally quantified • Here, c is a skolem variable while f is a skolem function

11. Example

12. Herbrand Universe • It is infeasible to consider all interpretations over all domains in order to prove unsatisfiability • Instead, we try to fix a special domain (called a Herbrand universe) such that the formula, S, is unsatisfiable iff it is false under all the interpretations over this domain

13. Herbrand Universe (contd.) • H0 is the set of all constants in a set of clauses, S • If there are no constants in S, then H0 will have a single constant, say H0 = {a} • For i=1,2,3,…, let Hi+1 be the union of Hi and set of all terms of the form fn(t1,…, tn) for all n-place functions f in S, where tj where j=1,…,n are members of the set H • H∞ is called the Herbrand universe of S

14. Herbrand Universe (contd.) • Atom Set: Set of the ground atoms of the form Pn(t1,…, tn) for all n-place predicates Pn occuring in S, where t1,…, tn are elements of the Herbrand Universe of S • Also called the Herbrand Base • A ground instance of a clause C of a set of clauses is a clause obtained by replacing variables in C by members of the Herbrand Universe of S

15. Example

16. H-Interpretations • For a set of clauses S with its Herbrand universe H, we define I as an H-Interpretation if: • I maps all constants in S to themselves • An n-place function f is assigned a function that maps (h1 ,…, hn) (an element in Hn) to f (h1 ,…, hn) (an element in H) where h1 ,…, hn are elements in H • Or simply stated as I={m1, m2, …, mn, …} where mj = Aj or ~Aj (i.e. Aj is set to true or false) and A = {A1, A2, …, An, …}

17. H-Interpretations (contd.) • Not all interpretations are H-Interpretations • Given an interpretation I over a domain D, an H-Interpretation I* corresponding to I is an H-Interpretation that: • Has each element from the Herbrand Universe mapped to some element of D • Truth value of P(h1 ,…, hn) in I* must be same as that of P(d1 ,…, dn) in I

18. Example

19. Use of H-Interpretations • If an interpretation I satisfies a set of clauses S, over some domain D, then any one of the H-Interpretations I* corresponding to I will also satisfy H • A set of clauses S is unsatisfiable iff S is false under all H-Interpretations of S

20. Semantic Trees • Finding a proof for a set of clauses is equivalent to generating a semantic tree • A semantic tree is a tree where each link is attached with a finite set of atoms or their negations, such that: • Each node has only a finite set of immediate links • For each node N, the union of sets connected to links of the branch down to N does not contain a complementary pair • If N is an inner node, then its outgoing links are marked with complementary literals

21. Semantic Trees (Contd.) • Every path to a node N does not contain complementary literals in I(N), where I(N) is the set of literals along the edges of the path • A Complete Semantic Tree is one in which every path contains every literal in Herbrand base either +ve or –ve, but not both • A failure node N is one which falsifies IN but not IN’, where N’ is predecessor of N • A semantic tree is closed if every path contains a failure node

22. Example S’ is satisfiable because it has at least one branch without a failure node Image courtesy: http://www.computational-logic.org/iccl/master/lectures/summer07/sat/slides/semantictrees.pdf

23. Example S is unsatisfiable as the tree is closed Image courtesy: http://www.computational-logic.org/iccl/master/lectures/summer07/sat/slides/semantictrees.pdf

24. Herbrand’s Theorem (Ver. 1) Theorem: A set S of clauses is unsatisfiable iff corresponding to every complete semantic tree of S, there is a finite closed semantic tree Proof: Part 1: Assume S is unsatisfiable - Let T be the complete semantic tree for S - For every branch B of T, we let IB be the set of all literals attached to the links in B

25. Version 1 Proof (contd.) - IB is an interpretation of S (by definition) - As S is unsatisfiable, IB must falsify a ground instance of a clause C in S, let’s call it C’ - T is complete, so, C’ must be finite and there must exist a failure node NB (a finite distance from root) on branch B - Every branch of T has a failure node, so we find a closed semantic tree T’ for S - T’ has a finite no. of nodes (Konig’s Lemma) Hence, first half of thm. is proved

26. Version 1 Proof (contd.) Part 2: If there is a finite closed semantic tree for every complete semantic tree of S - Then every branch contains a failure node - i.e. every interpretation falsifies S - Hence, S is unsatisfiable Thus, both halves of the theorem are proved

27. Herbrand’s Theorem (Ver. 2) Theorem: A set S of clauses is unsatisfiable iff there is a finite unsatisfiable set S’ of ground instances of clauses of S Proof: Part 1: Assume S is unsatisfiable - Let T be a complete semantic tree of S - By ver. 1 of Herbrand Thm., there is a finite closed semantic tree T’ corresponding to T

28. Version 2 Proof (contd.) - Let S’ be a set of all the ground instances of clauses that are falsified at all failure nodes of T’ - S’ is finite since T’ contains a finite no. of failure nodes - Since S’ is false in every interpretation of S’, S’ is also unsatisfiable Hence first half of thm. is proved

29. Version 2 Proof (contd.) Part 2: Suppose S’ is a finite unsatisfiable set of gr. instances of clauses in S - Every interpretation I of S contains an interpretation I’ of S’ - So, if I’ falsifies S’, then I must also falsify S’ - Since S’ is falsified by every interpretation I’, it must also be falsified by every interpretation I of S - i.e. S is falsified by every interpretation of S - Hence S is unsatisfiable Thus, both halves of the thm. are proved

30. Example

31. References • Chang, Chin-Liang and Lee, Richard Char-TungSymbolic Logic and Mechanical Theorem ProvingAcademic Press, New York, NY, 1973